Optimal frequency assignments of cycles and powers of cycles

Byline: Hsun-Wen Chang, Hao-Wen Chen To reduce the interfering among transmitters, any two 'close' transmitters, with distance no more than d2, must receive frequencies at least l2 apart, and any two 'very close' transmitters, with distance no more than d1, must receive channels at least l1 apart. The assignment problem is to assign a frequency to each transmitter such that total bandwidth of assigned frequencies can be minimised. In this paper, we present upper and lower bounds of the minimum bandwidth of cycles and powers of cycles. We also show the corresponding assignment functions. And then we partially solve the assignment problem of cycles Cn and their powers such as: (i) 3 ≤ n ≤ 2d1+1; (ii) 2d1 + 2 ≤ n ≤ 2d2 + 1, q1 ≠ 0 and l1 ≤ m1l2; (iii) 2d1 + 2 ≤ n ≤ 2d2 + 1, q1 ≡ 1(mod m1) and l1 > m1l2; (iv) 2d1 + 2 ≤ n ≤ 2d2 + 1, q1 = 0 and l1 ≥ (m1 + d1 − 1)l2 or l1 ≤ (m1 − 1)l2; (v) n ≥ 2d2 + 2 and l1 ≤ ⌊(d2 + ⌊q2/m2⌋)/(d1 + 1)⌋l2; and (vi) n ≥ 2d2 + 2, q1 ≡ 1(mod m1), and l1 > m1l2, where n = mi(di + 1) + qi and 0 ≤ qi ≤ di.